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pdiv_inc()

Polynomial division with increasing terms powers
(226 downloads for this version - 372 downloads for all versions)
Details
Version
2.0
Author
Samuel Gougeon
Maintainer
Samuel Gougeon
Category
License
Supported Scilab Version
5.5
Creation Date
August 27, 2015
Description
            --> pdiv_inc

 Polynomial division with increasing terms powers

 SYNTAX
 ------
          pdiv_inc   // Displays this help
 [q, r] = pdiv_inc(p, d)
 [q, r] = pdiv_inc(f)

 PARAMETERS
 ----------
 p,d,q,r: Single polynomials with real or complex coefficients.
          p : polynomial to be divided
          d : divisor. Its first non-null coefficient of lowest power must
              be of power equal or lower than p's one.
          q : quotient, with deg(q) <= deg(p)
          r : remainder, with deg(r) <= deg(p)+deg(d)
          such that: p = q*d + r. p and d are not necessarily coprime.

 f : Single rational, with coprime polynomials p = numer(f) and d = denom(f)

 DESCRIPTION
 -----------
 pdiv_inc(p, d) performs the Euclidian division p/d with polynomial terms
 of p and d sorted by increasing powers. It returns the quotient q and the
 remainder r, such that p = q*d + r.

 Detailled example:
   2 - x + 3x^2               | 1 + 2x^2
   ---------------------------|------------
   2     + 4x^2               | 2 - x - x^2
     - x -  x^2               |
     - x        - 2x^3        |
         -  x^2 + 2x^3        |
         -  x^2        - 2x^4 |
                  2x^3 + 2x^4 |
  Hence, (2 - x + 3x^2) = (2 - x - x^2)*(1 + 2x^2) +  (2x^3 + 2x^4)

 NOTES:
  * If d==0, q=0 and r=p are returned whatever is p.
  * if both p and d are limited to a decimal, there are set to p+0*x and d+0*x

 WARNING: If f is specified explicitly through a p/d ratio with p and d not
  coprime, their common factors are first simplified, and the p = q*d + r 
  relationshift holds only with simplified p => sp and d => sd.
  Example: p = poly([1 2 2 3 3],'x'), d = poly([2 3 4],'x'), p/d
          [q,r] = pdiv_inc(p/d), p-q*d-r, f = p/d; numer(f)-q*denom(f)-r

 REFERENCE
 ---------
  Comments, scoring and bug reports are welcome on
  http://fileexchange.scilab.org/toolboxes/449000#new_comment

 SEE ALSO
 --------
  pdiv   : Polynomial division by decreasing terms powers
  modulo : Remainder after polynomial division
  ldiv   : Limited Taylor series of a polynomial ratio

 EXAMPLES
 --------
 x = poly(0,"x");
// 1) Simple example
 p =  1 + 3*x + 2*x^2 - 7*x^3
 d = -1 +   x - 2*x^2
 [q,r] = pdiv_inc(p,d),  p-q*d-r   // => 0

// 2) With some non-leading null coefficients in p and/or d
 p = 1 + x +      3*x^3
 d = 1 +    2*x^2
 [q,r] = pdiv_inc(p,d),  p-q*d-r   // => 0

// 3) With p not starting with x^0
 p =    x -   x^2 + 2*x^3
 d = -1   + 2*x^2
 [q,r] = pdiv_inc(p,d),  p-q*d-r   // => 0

// 4) With p and d not starting with x^0
 p =       x^2 + 2*x^3
 d =  -x + 2*x^2 - x^3
 [q,r] = pdiv_inc(p,d),  p-q*d-r   // => 0

// 5) with complex coefficients:
 p = -1 + (1+%i)*x - %i*x^2 ;
 d = %i - x + (1-%i)*x^2    ;
 [q,r] = pdiv_inc(p,d);
 sci2exp(q), sci2exp(r), sci2exp(p-q*d-r)   // => 0

// 6) d must start at a degree lower or equal to p's starting degree
 p =  2 -  x + 3*x^2
 d =     2*x -   x^2
 [q,r] = pdiv_inc(p,d)  //=> Error

// 7) Random example
 p = poly(grand(1,5,"uin",-9,9), "x", "coeff")
 d = poly(grand(1,7,"uin",-9,9), "x", "coeff")
 [q,r] = pdiv_inc(p,d),  clean(p-q*d-r)   // => 0
            
Files (2)
[12.18 kB]
Screenshot
Screenshot of an example for thumbnail
[6.36 kB]
Miscellaneous file
File defining the function pdiv_inc(), help contents and examples

* MAKE pdiv_inc() AVAILABLE in EVERY SESSION: Click here to see HOW TO

* Get help: Enter "pdiv_inc" without parameters

CHANGES 1.0 => 2.0
------------------
 * Minor improvements in the help
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